Understanding the Exponential Equation

We are presented with an exponential equation involving the base $\frac{5}{3}$. The equation is given as:

$(\frac{5}{3})^{x-3} = \frac{3125}{243}$

The main objective is to determine the value of '$x$' first, and subsequently calculate the value of the expression $(\frac{5}{3})^{\frac{x}{2}}$.

Simplifying the Given Equation to Find '$x$'

To solve for '$x$', it's essential to express both sides of the equation using the same base. The left side already has the base $\frac{5}{3}$. We need to rewrite the right side, $\frac{3125}{243}$, as a power of $\frac{5}{3}$.

• Let's find the powers of 5 and 3:
  • $5^1 = 5$, $5^2 = 25$, $5^3 = 125$, $5^4 = 625$, $5^5 = 3125$.
  • $3^1 = 3$, $3^2 = 9$, $3^3 = 27$, $3^4 = 81$, $3^5 = 243$.
• From this, we can see that $3125 = 5^5$ and $243 = 3^5$.
• Therefore, we can write $\frac{3125}{243}$ as $\frac{5^5}{3^5}$.
• Using the property of exponents, $(\frac{a}{b})^n = \frac{a^n}{b^n}$, we get $\frac{5^5}{3^5} = (\frac{5}{3})^5$.

Substituting this back into the original equation gives us:

$(\frac{5}{3})^{x-3} = (\frac{5}{3})^5$

Because the bases on both sides of the equation are identical ($\frac{5}{3}$), their exponents must be equal:

$x-3 = 5$

To isolate '$x$', we add 3 to both sides of the equation:

$x = 5 + 3$

$x = 8$

Calculating the Value of $(\frac{5}{3})^{\frac{x}{2}}$

Now that we have determined that $x=8$, we can proceed to calculate the value of the target expression $(\frac{5}{3})^{\frac{x}{2}}$.

Substitute the value $x=8$ into the expression:

$(\frac{5}{3})^{\frac{8}{2}}$

First, simplify the exponent $\frac{8}{2}$:

$(\frac{5}{3})^4$

Finally, calculate the value of $(\frac{5}{3})^4$:

• Calculate the numerator: $5^4 = 625$.
• Calculate the denominator: $3^4 = 81$.

So, the value of the expression is:

$(\frac{5}{3})^4 = \frac{625}{81}$

The value of $(\frac{5}{3})^{\frac{x}{2}}$ is $\frac{625}{81}$.